Prime Number and Prime Factorization Calculator

What is a prime number and what does this tool do?

A prime number is a whole number greater than 1 that is divisible only by 1 and by itself: 2, 3, 5, 7, 11, 13 and so on. This tool tells you whether the number you type is prime. If it is not, it splits the number into prime factors, lists every positive divisor, gives the number and sum of the divisors and finds the nearest prime below and above. The second tab lists all primes between two bounds.

How do you check that a number is prime?

Divide n by each prime that does not exceed its square root. If none of them divides evenly, n is prime. Stopping at the square root is enough, because if n = a · b then one of the two factors is at most the square root. The tool shows these divisions with their remainders step by step.

Example

For 97 the square root is about 9.8, so the primes to try are 2, 3, 5 and 7. The remainders are 1, 1, 2 and 6. None is zero, so 97 is prime. 360 is not prime: it factors as 2³ · 3² · 5. Add one to each exponent and multiply to count the divisors: (3 + 1) · (2 + 1) · (1 + 1) = 24.

Primes in a range

The sieve of Eratosthenes starts at 2 and crosses out the multiples of each prime. Whatever is left is prime. There are 25 primes from 1 to 100 and 168 from 1 to 1000.

Limits

The single number check works up to 2⁶⁴ − 1 and the answer is exact up to that limit. The range scan goes up to 1 000 000. Type numbers with digits only. Spaces between digit groups are fine. Primality is not defined for decimals or negative numbers.